Seismic data interpolation beyond aliasing using regularized nonstationary autoregression

Field Data Examples

We use a set of marine 2-D shot gathers from a deepwater Gulf of Mexico survey (Crawley et al., 1999; Fomel, 2002) to further test the proposed method. Figure 5 shows the data before and after subsampling in the offset direction. The shot gather has long-period multiples and complicated diffraction events caused by a salt body. Amplitudes of the events are not uniformly distributed. Subsampling by a factor of 2 (Figure 5b) causes visible aliasing of the steeply dipping events. We designed a nonstationary PEF, with 15 (time) $\times$ 5 (space) coefficients for each sample and a 50-sample (time) $\times$ 20-sample (space) smoothing radius to handle the variability of events. Figure 6 shows the interpolation result and the difference between interpolated traces and original traces plotted at the same clip value. The proposed method succeeds in the sense that it is hard to distinguish interpolated traces from the interpolation result alone. A close-up comparison between the original and interpolated traces (Figure 7) shows some small imperfections. Some energy of the steepest events is partly missing. Coefficients of the adaptive PEF are illustrated in Figure 8, which displays the first coefficient ($B_1$ ) and the mean coefficient of $B_n$ , respectively. The filter coefficients vary in time and space according to the curved events. The interpolated results are relatively insensitive to the smoothing parameters.

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Figure 5. A 2-D marine shot gather. Original input (a) and input subsampled by a factor of 2 (b).

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Figure 6. Shot gather after trace interpolation (adaptive PEF with 15 $\times$ 5) (a) and difference between original gather (Figure 5a) and interpolated result (Figure 6a) (b).

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Figure 7. Close-up comparison of original data (a) and interpolated result by RNA (b).

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Figure 8. Adaptive PEF coefficients. First coefficient $B_1$ (a) and mean coefficient of $B_n$ (b).

For a missing-trace interpolation test (Figure 9a), we removed 40% of randomly selected traces from the input data (Figure 5a). Furthermore, the first five traces were also removed to simulate traces missing at near offset. The adaptive PEF can only use a small number of coefficients in the spatial direction because of a small number of fitting equations (where the adaptive PEF lies entirely on known data). However, it also limits the ability of the proposed method to interpolate dipping events. We used a nonstationary PEF with 4 (time) $\times$ 3 (space) coefficients for each sample and a 50-sample (time) $\times$ 10-sample (space) smoothing radius to handle the missing trace recovery. The result is shown in Figure 9b. By comparing the results with the original input (Figure 5a), the missing traces are interpolated reasonably well except for weaker amplitude of the steeply dipping events.

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Figure 9. Field data with 40% randomly missing traces (a), and reconstructed data using RNA (b).

An extension of the method to 3-D is straightforward and follows a two-step least-squares method with 3-D adaptive PEF estimation. We use a set of shot gathers as the input data volume to further test our method (Figure 10a). We removed 50% of randomly selected traces and five near offset traces for all shots (Figure 10b). For comparison, we used PWD to recover the missing traces (Figure 11a). The PWD method produces a reasonable result after carefully estimating dip information, but the interpolated error is slightly larger in the diffraction locations. (Figure 11b). The additional direction provided more information for interpolation but also increased the number of zeros in the mask operator $K(t,x)$ , which constrains enough fitting equations in equation 13. To use the available fitting equations for adaptive PEF estimation, we chose a smaller number of coefficients in the spatial direction. The proposed method is able to handle conflicting dips, although it does not appear to improve the dipping-event recovery compared to the 2-D case. This characteristic partly limits the application of RNA in 3-D case. We used a 3-D nonstationary PEF with 4 (time) $\times$ 2 (space) $\times$ 2 (space) coefficients for each sample and a 50-sample (time) $\times$ 10-sample (space) $\times$ 10-sample (space) smoothing radius was selected. Similar to the result in the 2-D example, Figure 11c shows the interpolation result, in which only steeply-dipping low-amplitude diffraction events with are lost (Figure 11d).

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Figure 10. A 3-D field data volume (a) and data with 50% randomly missing traces (b).

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Figure 11. Reconstructed data volume using 3-D plane-wave destruction (a), difference between original input (Figure 10a) and interpolated result (Figure 11a), reconstructed data volume using 3-D regularized nonstationary autogression (c), and difference between original input (Figure 10a) and interpolated result (Figure 11c) (d).

Seismic data interpolation beyond aliasing using regularized nonstationary autoregression